Revista Matemática Iberoamericana

$m$-Berezin transform and compact operators

Kyesook Nam , Dechao Zheng , and Changyong Zhong

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$m$-Berezin transforms are introduced for bounded operators on the Bergman space of the unit ball. The norm of the $m$-Berezin transform as a linear operator from the space of bounded operators to $L^{\infty}$ is found. We show that the $m$-Berezin transforms are commuting with each other and Lipschitz with respect to the pseudo-hyperbolic distance on the unit ball. Using the $m$-Berezin transforms we show that a radial operator in the Toeplitz algebra is compact iff its Berezin transform vanishes on the boundary of the unit ball.

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Rev. Mat. Iberoamericana, Volume 22, Number 3 (2006), 867-892.

First available in Project Euclid: 22 January 2007

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Zentralblatt MATH identifier

Primary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15] 47B38: Operators on function spaces (general)

$m$-Berezin transforms Toeplitz operators


Nam, Kyesook; Zheng, Dechao; Zhong, Changyong. $m$-Berezin transform and compact operators. Rev. Mat. Iberoamericana 22 (2006), no. 3, 867--892.

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  • Ahern, P., Flores, M. and Rudin, W.: An invariant volume-mean-value property. J. Funct. Anal. 111 (1993), 380-397.
  • Axler, S. and Zheng, D.: The Berezin transform on the Toeplitz algebra. Studia Math. 127 (1998), 113-136.
  • Axler, S. and Zheng, D.: Compact operators via the Berezin transform. Indiana Univ. Math. J. 47 (1998), 387-399.
  • Békollé, D., Berger, C. A., Coburn, L. A. and Zhu, K. H.: BMO in the Bergman metric on bounded symmetric domains. J. Funct. Anal. 93 (1990), 310-350.
  • Berezin, F. A.: Covariant and contravariant symbols of operators. Math. USSR-Izv. 6 (1972), 1117-1151.
  • Berger, C. A. and Coburn, L. A.: Toeplitz operators on the Segal-Bargmann space. Trans. Amer. Math. Soc. 301 (1987), 813-829.
  • Berger, C. A. and Coburn, L. A.: Heat flow and Berezin-Toeplitz estimates. Amer. J. Math. 116 (1994), 563-590.
  • Bratteli, O. and Robinson, D. W.: Operator algebras and quantum statistical mechanics. I, II. Texts and Monographs in Physics. Springer Verlag, New York 1979, 1981.
  • Choe, B. R. and Lee, Y. J.: Pluriharmonic symbols of commuting Toeplitz operators. Illinois J. Math. 37 (1993), 424-436.
  • Coburn, L. A.: A Lipschitz estimate for Berezin's operator calculus. Proc. Amer. Math. Soc. 133 (2005), 127-131.
  • Engliš, M.: Compact Toeplitz operators via the Berezin transform on bounded symmetric domains. Integral Equations Operator Theory 33 (1999), 426-455.
  • Gohberg, I. C. and Kreĭ n, M. G.: Introduction to the theory of linear nonselfadjoint operators. Translations of Mathematical Monographs 18. American Mathematical Society, Providence, R.I., 1969.
  • Miao, J. and Zheng, D.: Compact operators on Bergman spaces. Integral Equations Operator Theory 48 (2004), 61-79.
  • Rudin, W.: Function Theory in the unit ball of $\mathbbC^n$. Fundamental Principles of Mathematical Science 241. Springer-Verlag, New York-Berlin, 1980.
  • Stroethoff, K.: The Berezin transform and operators on spaces of analytic functions. In Linear operators (Warsaw, 1994), 361-380. Banach Center Publ. 38. Polish Acad. Sci., Warsaw, 1997.
  • Suárez, D.: Approximation and symbolic calculus for Toeplitz algebras on the Bergman space. Rev. Mat. Iberoamericana 20 (2004), 563-610.
  • Suárez, D.: Approximation and the $n$-Berezin transform of operators on the Bergman space. J. Reine Angew. Math. 581 (2005), 175-192.
  • Zygmund, A.: Trigonometric series. Cambridge University Press, New York, 1959.